or (REF ) when written in terms of the Green's function.
This heuristic derivation can be made rigorous as shown in [1]}.
This equation is written in Stratonovich form as represented by the \(\circ \)  symbol.
This reflects the fact that we arrive at this description as a scaling limit of a physical model where the fluctuating random field \(\nu \)  multiplying
the wave field \(u\)  has a finite correlation length in the \(z\) -direction.
The Stratonovich stochastic integral can be interpreted in the simplest case as
the limit when the integrand is evaluated at the midpoint of the interval of increment
of the driving Brownian field and thus naturally appears in the diffusion
limit when \(\nu \)  is replaced by a driving Brownian field.
Note that the Itô interpretation
of (REF ) has the form
\(d v = \frac{i}{2k}\Delta _{{\itbf x}}v \, dz - \frac{k^2 C({\bf 0})}{8} v dz+ \frac{i k}{2 } v \, dB(z,{{\itbf x}}) .\) 
